All functions mentioned in this paper will be real-valued. If f
2, g are nonnegative functions on a set S that satisfy g ≦ f
1 + f
2, the Riesz decomposition problem associated with these data is to find functions gi
on S such that
always furnishes a solution. The problem becomes more interesting if one asks under what conditions one can find solutions that are, roughly speaking, as smooth as the data.